Math Final

Front 1

Minimum

Back 1

m<= x for every x in S

Front 2

Well ordering principle for N

Back 2

every non-empty subset of N has a minimum

Front 3

Even number

Back 3

n in N is even if n = 2k for some k in N

Front 4

Odd number

Back 4

n is odd if n=2k-1 for some k in N

Front 5

Additive identity

Back 5

z is an additive identity for number system if n+z=z+n=n for every n

Front 6

Multiplicative identity

Back 6

u is a multiplicative identity if n*u=u*n=n, for every n

Front 7

Additive inverse

Back 7

b is an additive inverse of a if a+b=b+a=0

Front 8

Multiplicative inverse

Back 8

b is a multiplicative inverse of a if a*b=b*a=1

Front 9

Q

Back 9

Q={p/q | p in Z, g in N} C F

Front 10

m/n

Back 10

[m//n] = {m/n | p//q = m//n}

Front 11

|x|

Back 11

= { 1. x if x>= 0

{ 2. -x if x < 0

Front 12

R

Back 12

a complete ordered field

Front 13

Completeness Axiom

Back 13

say an ordered field is complete when every non empty subset of F, which is bounded below, has an infimum in F

Front 14

Archimedean principle

Back 14

If r in R, there exists n in N such that 1/n < E (i.e. for every E>0, there exists n in N such that 1/n < E)

Front 15

Density Theorem

Back 15

Let a,b, in R with a<b, Then there exists r in Q such that a<r<b

Front 16

Hypothesis & Sufficient Condition

Back 16

P

Front 17

Conclusion & Necessary Condition

Back 17

Q

Front 18

Contrapositive

Back 18

(~Q) -> (~P)

Front 19

Tautology

Back 19

A statement that's always true

Front 20

Contradiction

Back 20

A statement that's always false

Front 21

Subset

Back 21

A is a subset of B when if x in A, then x in B

Front 22

Equals

Back 22

A equals B when A is a subset of B and B is a subset of A

Front 23

A \ B

Back 23

{x in A and x not in B}

Front 24

Union

Back 24

{x| there exists i in I s.t. x is in Si}

Front 25

Intersection

Back 25

{x| for every i in I, x is in Si}

Front 26

Ordered pair

Back 26

a set of the form {{a}, {a,b}}

Front 27

Cartesian Product

Back 27

{(x,y) | x in A and y in B}

Front 28

Relation

Back 28

a relation between A and B is any subset of AxB

Front 29

Relation on A

Back 29

a subset of A x A

Front 30

Total order

Back 30

A relation is total order if it is transitive and has trichotomy

Front 31

Lower bound

Back 31

If A C U, we say m in U is a lower bound of A when if x in A, then m<=x

Front 32

Upper bound

Back 32

If A C U, we say m in U is an upper bound of A when if x in A, then x<=m

Front 33

Minimum

Back 33

If A C U, we say m in U is a minimum of A when if x in A, then m<=x; and m in A

Front 34

Maximum

Back 34

If A C U, we say m in U is a maximum of A when if x in A, then x<=m; and m in A

Front 35

Infimum

Back 35

If A C U, we say m in U is an infimum of A when

If x in A, then m <=x &

If l > m, then there exists x in A s.t. X < l

Front 36

Supremum

Back 36

If A C U, we say m in U is a supremum of A when

If x in A, then x <= m &

If l < m, then there exists x in A s.t. x>l

Front 37

Well ordered

Back 37

Let U be a set with total order. We say U is well ordered when, if A C is not empty then A has a minimum

Front 38

Complete

Back 38

Let U be a set with total order. We say that U is complete when, if A C U is not empty and is bounded below, then there exists m in U s.t. M = inf(A)

Front 39

Equivalence Relation

Back 39

Let A be a set. A relation on A is an equivalence relation when it is reflexive, symmetric and transitive.

Front 40

Equivalence Class

Back 40

Let A be a set with an equivalence relation. For any a in A, the equivalence class of a is the set [a] = {x in A| x == a}

Front 41

Modulo Equivalence A/==

Back 41

{[a]| a in A}

Front 42

Function

Back 42

a triple (f,A,B) where f C A*B such that

If a in A, then there is b in B so that (a,b) is in f &

If (a,b1) is in f and (a,b2) is in f, then b1 = b2

Front 43

Codomain

Back 43

B

Front 44

Composition

Back 44

Composite of f and g is a function g o f: A -> C defined by (g o f)(x) = g(f(x))

Front 45

Range(f)

Back 45

{b in B | there exists a in A such that (a,b) is in f}

Front 46

Injective

Back 46

when if f(a1) = f(a2), then a1 = a2

Front 47

Inverse

Back 47

g is an inverse of f when

(g o f)(x) = ia(x) &

(f o g)(x) = ib(x)

Front 48

Surjective

Back 48

if b is in B, then there exists a in A such that f(a) = b

Front 49

Bijective

Back 49

When f is injective and surjective

Front 50

Image

Back 50

Let f:A->B and S C A. image of S under f(x) is f(S) = {y in B| there exists x in S s.t. f(x)=y}

Front 51

Preimage

Back 51

preimage of T under f(x) is f-1(t)={x in A| f(x) is in T}

Front 52

Finite set

Back 52

A is a set and finite when if f:A->A is an injective function with domain A, then f(x) is surjective

Front 53

Infinite set

Back 53

A is a set and infinite when there exists f:A->A an injective function with doain A where f(x) is not surjective

Front 54

Countable

Back 54

A is countable if it is finite and denumerable

Front 55

Equinumerous

Back 55

If there exists a bijection f:A->B

Front 56

Denumerable

Back 56

If there exists a bijection f:N->A

Front 57

Uncountable

Back 57

If it is not countable